Relative Perturbation Theory for Quadratic Hermitian Eigenvalue Problem

Abstract

In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $\lambda^2 M x + \lambda C x + K x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices and $C$ is a general Hermitian matrix. We base our findings on new results for an equivalent regular Hermitian matrix pair $A-\lambda B$. The new bounds can be applied to many interesting quadratic eigenvalue problems appearing in applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments.